SU(3)XSU(2)XU(1)

This is the formulation of the Standard Model as a gauge theory with the gauge group SU(3)XSU(2)XU(1) or with a couple of fermion fields and a Higgs field, which is a and/or a . SU(3) describes Quantum chromodynamics, SU(2) describes the weak interaction* and U(1) describes hypercharge.

*Technically speaking, the Z boson is described by a field which is really a linear combination of SU(2) and U(1). See electroweak.

There are three families of fermions, each consisting of the representations, (q for left-handed quark), (d^c for the left-handed anti d-quark), (u^c for the left handed up antiquark), (l for the left handed leptons), (e^c for the left-handed positron) and (ν^c for the left-handed antineutrino, which is now known to exist. See Neutrino oscillation.).

The Higgs field acquires a VEV, resulting in a spontaneous symmetry breaking from or to .

Of course, calling the representations things like is purely a physicist's convention, not a mathematician's convention, where representations are either labelled by Young tableaux or Dynkin diagrams with numbers on their vertices, but still, it is standard among high energy physicists.

Since the homotopy group

this model no predicts monopoles associated with the electroweak breaking scale. See Hooft-Polyakov monopole.

The Yukawa couplings of the scalar Higgs fields with the fermions produces the fermion masses after the Higgs field acquires a VEV.